How do I fix vertices that aren't following the armature?
In Object Mode, select the character mesh. Go into Edit mode and makes sure the Properties Shelf is exposed (to the right of your 3D View). If it is not, press n to open it, so that you can see what happens when you select various vertices. Now select any of the vertices which is currently behaving properly. Immediately, you should see an entry appear in the Properties Shelf called Vertex Weights, which will have vertex group names listed, and their associated values.

Now select the vertex that's refusing to behave. You will likely discover that no Vertex Weights section appears in the Properties Panel when this vertex is selected. That would be because it is not assigned to any vertex groups currently, which would explain why it doesn't move when the bone moves.
To fix this, while still in Edit Mode, go to the Object Data tab in the Properties Panel, and then the Vertex Groups Section. Locate the vertex group in the list which you would like the problem vertex to become part of. Select the desired vertex group in the list of such groups, and then press the Assign button. Note the Weight bar. You may not want the vertex to have a weight of 1.0, which it will usually get when pressing the Assign button, because a weight of 1.0 is usually listed by default. Wherever you set this bar is what weight the vertex will get once assigned.

Once the offending vertex has been added to a vertex group that the bone already controls, and it has a level of influence greater than zero, the bone will control it. To further refine the influence, you can either use Weight Paint mode, or you can look in the Properties Shelf again for the Vertex Weights section, and now you will see the weight values listed for the newly added vertex.
One nice thing about this list view is that it displays every source of influence for the currently selected vertex, not merely one single source. You can manually adjust the level of influence for each source, or even remove a source altogether, using this display.